Yes, with tempo plugin you can monitor for yourself and for each individual in your team, the amount of time spent per project.
Here is the link to tempo plugin
Groovy can also do this for you and display against a custom field and then be used on a dashboard or view issue screen it's a little complex so I'd suggest building a plugin if tempo is over kill. Plus you can't display total time spent per day using tempo against a custom field only via a report (on screen/export). Though you could use the export function from tempo and use a plugin like nFeed to look at the report and display the data back into JIRA against and custom field..
In fact, the plugin has two templates that solve 80% of this, and you can achieve the remaining 20% with reconfiguring the pivot tables in those.
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